Bessel's inequality
Suppose e_1, e_2, ... is an orthonormal system (each a unit vector, all mutually perpendicular). The numbers <v, e_k> are the coordinates of v along each direction — its Fourier coefficients. Bessel's inequality says these coordinates can never overshoot the vector: sum_k |<v, e_k>|^2 <= ||v||^2.
The proof is one line of Pythagoras. Let p = sum_k <v, e_k> e_k be the projection of v onto the span of the system. Then v = p + (v - p) with v - p orthogonal to p, so ||v||^2 = ||p||^2 + ||v - p||^2 >= ||p||^2 = sum |<v, e_k>|^2. The gap is exactly the squared length of the unrepresented remainder.
Equality, sum_k |<v, e_k>|^2 = ||v||^2, holds precisely when the remainder vanishes — when the orthonormal system is COMPLETE (an orthonormal basis) so it captures all of v. That sharpened equality is Parseval's identity, and it says no energy leaks outside the coordinate directions.
This is the workhorse of Fourier analysis: it guarantees that infinite sums of squared Fourier coefficients converge (bounded by ||v||^2), which is the first step to making infinite-dimensional expansions rigorous. Bessel bounds; Parseval, with completeness, achieves the bound.
Squared Fourier coefficients are bounded by the squared norm; the inequality becomes Parseval's equality exactly when the system is a basis.
The slack in Bessel is exactly ||v - p||^2, the squared distance from v to the span of your system. Closing that gap to zero is precisely what completeness (a basis) means.